B Element half life "time"

1. Jul 2, 2017

girts

Just curious , have there been any such circumstances in the history of this universe (i'm thinking close or near the big bang moment of hot dense plasma) or are there possible such a scenario even theoretically where the half lives of isotopes and elements change?
In other words for example Cs137 (probably one of the most popular isotopes in the popular science) has a half life of 30 years, are there or have there been or are there even possible any such circumstances under which this period could be shorter or longer or does half life is "written in stone" with respect to any other changes or states of matter one could possibly put an element into?

thanks.

2. Jul 2, 2017

Staff: Mentor

There are a few exotic cases where the environment can influence the decay properties. Three examples:

- Beryllium-7 can only decay via electron capture. Normally there are electrons around to capture, so it is radioactive. Remove all electrons from its environment and it cannot decay any more - it gets stable.
- A neutral dysprosium-163 atom is stable. If you remove all electrons, it can beta decay, where the electron stays in a low energy level. The energy is not sufficient to have the electron escape or occupy a higher energy level, which would be required for a decay of a neutral atom.
- In neutron stars, nuclei with a larger neutron to proton ratio can be stable, as a beta decay would have to emit an electron to a very high-energetic state, higher than in regular matter. For the same reason things stable on Earth can be radioactive inside neutron stars - they can decay via electron capture.

3. Jul 8, 2017

Aidan Davis

There are many cases of exotic decays.
All nuclei (except the proton) can undergo alpha, neutron, or proton emission, or even fission in some cases if excited to the point where binding energy has been overcome. For example, beryllium emits a neutron if struck by a photon above 1.665 MeV, and Gd 157 will emit an alpha particle with 7.093 MeV if hit by a neutron of any energy. Both of these elements are stable in the ground state, however.
Any isotope that would normally undergo electron capture decay is stable if fully ionized (ie, there are no electrons to capture)
In the sun, when two protons fuse, helium 2 is made. Normally, this would decay by proton emission, but in these conditions beta plus decay occurs.
The half lifes of radioisotopes are fixed when the element is in its ground state. Some, such as Be 8, have a metastable state. This state of Be 8 allows the triple alpha process to continue more easily, as ground state Be 8 decays in 8.19*10^-17 seconds.http://www.astro.princeton.edu/~gk/A403/fusion.pdf
Somewhere on the first few pages of this the triple alpha process is shown with the metastable state of Be 8, however this paper's overall topic is unrelated.

4. Jul 26, 2017

girts

Oh one more question that i thought about , i read that half life is exponential instead of being linear in nature does that mean that the element begins with say 100 unstable atoms and for example after a year has 50 so half life is one year but does it also mean that say 9 months into the decay the element will still be at 100 unstable atoms instead of say some linearly decreased number like 61 ? Because in water evaporation for example the amount of water decreases in a linear fashion but how is it with nuclear decay?

5. Jul 26, 2017

gleem

The atoms of a sample of radioactive material decay in a random manner. But on average a sample will decay to half its quantity in one half-life. If you had 100 samples of a 100 atoms you could not predict how many atoms would decay before the time of one half-life.for a given sample . Some might have more than 50 decays some less than 50 but if you averaged the number of decays for the 100 samples you would get close to 50. The more samples you had the closer you would get to 50.

On average the number of decays that you can expect to observe in a specific time is given by the equation

expected decays in time t= (Number of atoms that you start with)×(1/2)t/HL

6. Jul 26, 2017

Staff: Mentor

The decay of every individual atom is random. It can happen that all your atoms decay in the first day, or no atom decays in the first 100 years - it is just very unlikely.
On average, after time t, you expect that a fraction of $2^{.t/T}$ of the initial atoms is left, where T is the half-life. If you plug in T=1 year and t=1 year, you get 1/2, so on average 50% of the atoms are still in the initial state, or 50 out of your 100 atoms. After 9 months = 3/4 years, you expect 59 atoms to be without decay.
After 7 years, you expect that 0.78 atoms are left, which means there is a good probability that all 100 atoms decayed. After 12 years, you expect 0.025 remaining atoms, it is very likely that all of them decayed.

Edit: Wrote the reply in parallel to gleem.

7. Aug 1, 2017

girts

so are you then saying that half life is not very precise in terms of time it takes and that certain half life given in years is only made specific enough aka down to the precision of years or months simply looking from the experimental observation involving thousands of samples of radioactive material.
so theoretically we say that the half life of Cs137 for example is 30 years but it could very well be that if we only had a few samples of Cs137 around the world we could have very well defined the half life being 31 or 29 or otherwise years ?

So would it be fair to assume that radioactive half life is only precise due to the large number of example we have so we can calculate some given average out of all of them,

but then how precise is half life if the decay at any one given sample is random as you said?
does it then mean that the decay is kind of a large scale Schroedinger's cat? and that we can only know the approximate amount of atoms left if we have a large enough sample or many many samples but in any given one there might be a chance that half of the atoms only decay say a month before the end of its half life?

thanks.

8. Aug 1, 2017

Staff: Mentor

The half life is an exact value. The time where each atom decays is random, but if you have many atoms you'll see that their decays follow a fixed distribution - the more atoms you have the closer they will be to this theoretical distribution.

We cannot measure anything exactly, of course, there are always experimental uncertainties in measurements. The current most precise experimental result is 11018.3 ± 9.5 days.
No. All samples have the same half life. Our knowledge about this value is not perfect, but that is just an experimental limit.

9. Aug 1, 2017

rootone

Half life of an element is simply a statement of what is statistically true, given actual measurements.
As far as I know there is no theory which predicts what a half life should be, although some isotopes can be recognised as unstable, and so will be short lived.

10. Aug 1, 2017

Staff: Mentor

There are theoretical calculations of half lives, they are not very accurate but they exist.

11. Aug 2, 2017

girts

Ok, but then please let me be a bit ignorant here partly because I don't know and ask, if half life of an element is a parameter which we can only vaguely calculate based on theoretical assumptions but measure with much higher certainty in real life, then how do we know the half lives that are in the thousands and millions of years area?

the shorter half life materials are easy to prove since we have the technology to actually measure them for the last 80 or so years, but for example U235 which is a tiny portion of natural U has a said half life of 700 million years then how does one arrive at this conclusion scientifically because surely no one has been around for not even a thousand years constantly measuring the decay rate and then extrapolating that to an approximate higher number or otherwise?

Anyway the more I think about it the more it puzzles me, it is kind of interesting that if I got your answers correctly that in any given half life the atoms can decay in a non linear fashion either more at the beginning of the half life or at the end or otherwise but in the end of any given half life half of them would have decayed anyway,

the reason this is puzzling is because earlier I thought of half life as simply a linear fashion mechanism by which certain unstable atoms loose energy and settle down to more stable states, but here it seems almost like the atoms follow some kind of an "algorithm" because theoretically a given sample could have most of its atoms decayed well before the end of its half life yet still it waits patiently for that time to come and only then the full half number is reached even though it might have been just a few atoms away am I understanding this correctly, and only after that time the next half of what is left starts to decay ?

12. Aug 2, 2017

Staff Emeritus
There is a mathematical relation between a half-life and a quarter-life or three-quarters life or any other fraction you care to mention. For slowly decaying nuclei, one measures the time for 1% of them to decay (or even less) and converts.

13. Aug 2, 2017

vanhees71

Take the usual exponential decay law, according to which the probability distribution for a particle to have survived after time $t$ is
$$P(t)=A \exp(-\lambda t).$$
The constant $A$ is defined by the normalization condition
$$\int_0^{\infty} \mathrm{d} t P(t)=A \int_0^{\infty} \exp(-\lambda t) = \frac{A}{\lambda} \; \Rightarrow A=\lambda.$$
$$\tau=\int_0^{\infty} \mathrm{d}t P(t) t.$$
The integral can be evaluated by using the generating function
$$G(\lambda)=\int_0^{\infty} \mathrm{d} t \exp(-\lambda t)=\frac{1}{\lambda},$$
giving
$$\tau=-\lambda G'(\lambda)=\frac{1}{\lambda}.$$
Within the lifetime $\tau$ the survival probability has decreased by a factor $1/\mathrm{e}$.

To get the half-life time you need
$$\exp(-\lambda \tau_{1/2})=\frac{1}{2} \; \Rightarrow \; \tau_{1/2} =\frac{\ln 2}{\lambda}.$$
On average after this time half of a given unstable substance is decayed.

14. Aug 2, 2017

Staff: Mentor

There are many ways to measure the half life, and the measured values cover a range of more than 1050. They all use the mathematical relations between the time where an atom has 50% probability to decay (half life) and similar times for other probabilities.

For everything longer than a few years: Measure the initial number of atoms,. then measure the number of atoms that decay within a day, a month or similar experimentally accessible time frames. Mathematics relates this to the half life. The longest half life measured accurately occurs in Tellurium-128: 2.2*1024 years, or 7*1030 seconds, with an experimental uncertainty of about 10%.
For everything from minutes to several years: Prepare a sample, measure the activity over time and how it decreases.
For everything too short-living or too rare to prepare a sample: measure production time and decay time of individual particles.
For extremely short-living stuff there are some tricks based on quantum mechanics, but let's ignore that here.

15. Aug 3, 2017

vanhees71

Maybe a stupid question: How do you measure the extreme long lifetimes of $10^{24}$ years?

16. Aug 3, 2017

Staff: Mentor

Count the number of decays $\Delta N$ in a specified time $\Delta t$ in a sample of known size $N$.

$\frac {dN}{dt} = -N \lambda$ therefore $\Delta N \cong -N \lambda \Delta t$ is a very good approximation in such cases, where $\Delta t \ll t_{1/2}$.

17. Aug 3, 2017

Staff: Mentor

I also seem to remember reading about a method involving decay chains. Suppose isotope 1 with decay constant $\lambda_1$ decays into isotope 2 with decay constant $\lambda_2$ which in turn decays into isotope 3. If isotope 2 is in equilibrium, i.e. "new" nuclei are created by decay from isotope 1 at the same rate as are lost by decay to isotope 3, then $$\frac{dN_2}{dt} = \lambda_1 N_1 - \lambda_2 N_2 = 0$$ If you know $N_1$, $N_2$ and $\lambda_2$, you can calculate $\lambda_1$.

18. Aug 3, 2017

Staff: Mentor

In principle you could assemble 40 kg of Te-128 to get 100 decays per year, if you can detect all decays you get 10% statistical uncertainty after a year of measurement.
Searches for proton decays follow this approach with thousands of tons of matter.

In practice it is easier to measure the half-life of Te-130 (a different isotope, decays 2500 times faster), and to measure the ratio of half lives of Te-128 to Te-130 via very old rocks where you can count how many decays of both types happened. Here is a publication measuring the ratio, and one measuring Te-130.

19. Aug 4, 2017

girts

Thanks for the answers so far,

what I did not see mentioned here but I read it myself is that simply by measuring a tiny amount of time of decay for a very long half life element like U235 doesn't do anything because that would be like taking a small portion of a very long string and determining the length of the string simply by that portion which is impossible as there are too many unknowns but what I read is that for these long lived elements whose half life cannot be directly experimentally observed in any real time frame they simply make a precise measurement of the weight of the sample in question say 10 grams of U235 and by that they calculate the amount of atoms in that sample (assuming one can calculate the amount of atoms in a sample of known weigh with high certainty? please comment)

so technically by knowing how many atoms you have in your sample and measuring the rate of decay per given amount of time one then arrives at the conclusion for how long it will take for them to decay aka - half life.
I assume they also can reverse this calculation and come up with a number that the sample or U235 in general had billions of years ago in terms of how active it was back then?

Now let me ask the part which I don't quite get and would appreciate some detailed explanation.
In a lab frame surely they measure a tiny sample and don't bring in multi ton chunks of the material so assuming constant decay rate it would mean that in a sample that is small there are fewer atoms than in a sample that is large, so far so good, but since it is the same material it means that a certain amount of atoms will decay in a given time period but since the time period is so large it means that very few atoms decay in any real (human) amount of time, in a large sample like natural Uranium mines or otherwise there are much more atoms so more will decay in any given time but in a lab sample there are very few atoms compared to much bigger samples so from that tiny little sample maybe just one or two atoms will decay in the same amount of time correct?
so given that so few atoms decay and that there is a quantum uncertainty of atoms decaying which don't follow a linear relationship , how do we then can arrive at any precise or meaningful half life time for an element whose half life is said to be so large?
Do they arrive at a more precise conclusion simply by averaging over many samples and many tests ?

oh and also how could the half life been affected given that all those years ago the content of U235 was higher in natural Uranium and it was more radioactive or does this doesn't affect the matter?

thanks.

20. Aug 4, 2017

ofirg

What do you mean by that? What is and can be measured is the fraction of Nuclei that have decayed after a time T. Each Nuclei has either already decayed or not.

A few clarifications.

Each Nuclei decays independently at a time drawn from a distribution ( Exponential distribution). The half life is a parameter of this distribution which varies from Nuclei to Nuclei. As vanhees71 mentioned, The probability for a single nuclei to decay after time t is given by $\lambda exp(-\lambda t)$
where the half life is $\frac{ln(2)}{\lambda}$.

For measuring the half life accurately you need to observe a sample of Nuclei and measure how many have decayed at what times. Its important to stress that even if the have life is very large you still have a small probability that a given Nuclei will decay much sooner. The probability is about the ~elapsed time/half life ( times ln(2)). And since even a gram of material has many Nuclei if you can observe a small fraction which has decayed you can measure the half life in times much shorter than it.
you can combine different measurements of course, but there is no benefit in 'dividing' your sample. One large sample is as good as its sub samples. The total statistics is what matters.